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The given expression is:

\[
\frac{3x}{x+2} - \frac{4x}{2 - x} - \frac{2x - 1}{x^2 - 4}
\]

### Step-by-Step Simplification:

#### Step 1: Factor the denominator \( x^2 - 4 \)
We recognize that \( x^2 - 4 \) is a difference of squares:

\[
x^2 - 4 = (x + 2)(x - 2)
\]

So the third term becomes:

\[
\frac{2x - 1}{(x+2)(x-2)}
\]

#### Step 2: Rewrite the second fraction
We notice that the denominator \( 2 - x \) can be rewritten as \( -(x - 2) \). Thus:

\[
\frac{4x}{2 - x} = -\frac{4x}{x - 2}
\]

Now, the expression looks like this:

\[
\frac{3x}{x+2} + \frac{4x}{x-2} - \frac{2x - 1}{(x+2)(x-2)}
\]

#### Step 3: Find the common denominator
The least common denominator (LCD) is \( (x+2)(x-2) \). We now rewrite each term with this common denominator:

- The first term: 
\[
\frac{3x}{x+2} = \frac{3x(x - 2)}{(x+2)(x-2)} = \frac{3x^2 - 6x}{(x+2)(x-2)}
\]

- The second term: 
\[
\frac{4x}{x-2} = \frac{4x(x + 2)}{(x+2)(x-2)} = \frac{4x^2 + 8x}{(x+2)(x-2)}
\]

- The third term remains:
\[
\frac{-(2x - 1)}{(x+2)(x-2)}
\]

#### Step 4: Combine the numerators
Now, we combine all the terms over the common denominator:

\[
\frac{(3x^2 - 6x) + (4x^2 + 8x) - (2x - 1)}{(x+2)(x-2)}
\]

Simplifying the numerator:

\[
3x^2 - 6x + 4x^2 + 8x - 2x + 1 = 7x^2 + 0x + 1 = 7x^2 + 1
\]

#### Final Expression:

\[
\frac{7x^2 + 1}{(x+2)(x-2)}
\]

This is the simplified form of the given expression.

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Do you want more details or have any questions?

Here are 5 related questions for further practice:

1. How do you factor quadratic expressions like \( x^2 - 4 \)?
2. How do you rewrite fractions with negative denominators?
3. How do you determine the least common denominator (LCD) for rational expressions?
4. Can you find the domain of the simplified expression?
5. What steps are involved in combining rational expressions?

**Tip:** When combining fractions, always look for common factors and simplify step-by-step to avoid errors.

Simplify the expression 3x/(x+2) - 4x/(2-x) - (2x-1)/(x^2-4).

Learn how to simplify the rational expression 3x/(x+2) - 4x/(2-x) - (2x-1)/(x^2-4) using algebraic techniques such as factoring, rewriting denominators, and finding a common denominator. This problem is suitable for students in Grades 9-12 and covers key concepts like rational expressions and factoring.

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